scholarly journals The approach of a vortex pair to a plane surface in inviscid fluid

1979 ◽  
Vol 92 (3) ◽  
pp. 497-503 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Approximate Calculation ◽  
Plane Surface ◽  
Vortex Pair ◽  
Plane Wall ◽  
Inviscid Fluid ◽  
Viscous Effects ◽  
Axis Ratio ◽  
Local Rate

It is shown that a symmetrical vortex pair consisting of equal and opposite vortices approaching a plane wall at right angles must approach the wall monotonically in the absence of viscous effects. An approximate calculation is carried out for uniform vortices in which the vortices are assumed to be deformed into ellipses whose axis ratio is determined by the local rate of strain according to the results of Moore & Saffman (1971).

Viscosity Effects on the Radiation Hydrodynamics of Horizontal Cylinders

1991 ◽  
Vol 113 (4) ◽  
pp. 334-343 ◽  
Author(s):  
R. W. Yeung ◽  
C.-F. Wu
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Small Body ◽  
Low Frequency ◽  
Body Motion ◽  
Navier Stokes ◽  
Inviscid Fluid ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Horizontal Cylinders

The problem of a body oscillating in a viscous fluid with a free surface is examined. The Navier-Stokes equations and boundary conditions are linearized using the assumption of small body-motion to wavelength ratio. Generation and diffusion of vorticity, but not its convection, are accounted for. Rotational and irrotational Green functions for a divergent and a vorticity source are presented, with the effects of viscosity represented by a frequency Reynolds number Rσ = g2/νσ3. Numerical solutions for a pair of coupled integral equations are obtained for flows about a submerged cylinder, circular or square. Viscosity-modified added-mass and damping coefficients are developed as functions of frequency. It is found that as Rσ approaches infinity, inviscid-fluid results can be recovered. However, viscous effects are important in the low-frequency range, particularly when Rσ is smaller than O(104).

Stirring, stretching and transport generated by a pair of like-signed vortices

2011 ◽  
Vol 674 ◽  
pp. 244-280 ◽  
Author(s):  
F. RIZZI ◽  
L. CORTELEZZI
Keyword(s):  
Rotational Motion ◽  
Inner Core ◽  
Concentration Field ◽  
Vortex Pair ◽  
Chaotic Advection ◽  
Inviscid Fluid ◽  
Infinite Domain ◽  
Outer Flow ◽  
Initial Concentration ◽  
Angular Impulse

We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. We characterize this finite-time, aperiodic, dynamical system in terms of its fixed points and separatrices, which divide the flow into inner core, inner recirculation, outer recirculation regions and outer flow. We numerically simulate the time evolution of the vortex pair using a contour dynamics algorithm. The rotational and co-rotational motion of the vortices perturb the separatrices, which undergo to deformations, yielding a tangle whose complexity increases as the amplitude of the perturbation increases. We analyse the dynamics of the tangle and explain the transport of fluid between different regions. We use two diagnostics to quantify stirring: stretching of the interface and the mix-norm. These two diagnostics characterize stirring in contradicting ways and present different sensitivity to the parameters considered. We find that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mix-norm is dominated by the laminar transport induced within the outer recirculation regions. For pairs of vortices of small aspect ratio, stretching is piecewise linear and the mix-norm does not decrease monotonically. We show that these two effects are strongly coupled and synchronized with the rotational motion of the vortices. Since the nominal domain is unbounded, we quantify stirring on three concentric, circular domains. One domain nearly encloses the outer separatrices of the vortex pair, one is smaller and one larger than the first one. We show that the mix-norm is very sensitive to the size of the domain, while stretching is not. To quantify the sensitivity of stirring to the geometry of the initial concentration field, we consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation. We show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms.

scholarly journals Towards a new friction model for shallow water equations through an interactive viscous layer

2019 ◽  
Vol 53 (1) ◽  
pp. 269-299 ◽  
Author(s):  
François James ◽  
Pierre-Yves Lagrée ◽  
Minh H. Le ◽  
Mathilde Legrand
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Water Model ◽  
Finite Volume Scheme ◽  
Inviscid Fluid ◽  
Phase Lag ◽  
Viscous Layer ◽  
Viscous Effects ◽  
Shallow Water Models ◽  
Water Models

The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

Subsonic Compressible Flow Past Bluff Bodies

1954 ◽  
Vol 5 (3) ◽  
pp. 144-162 ◽  
Author(s):  
A. L. Longhorn
Keyword(s):  
Correction Factor ◽  
Fluid Velocity ◽  
Analytic Form ◽  
Prolate Spheroid ◽  
The Body ◽  
Inviscid Fluid ◽  
Bluff Bodies ◽  
Legendre Functions ◽  
Slender Body Theory ◽  
Axis Ratio

SummaryIn this paper the Janzen-Rayleigh method is used to calculate the velocity potential for the steady subsonic flow of a compressible, inviscid fluid past a prolate spheroid. The fluid velocity at a point on the body is calculated. The analytic form obtained for this velocity differs, from that giving the velocity which an incompressible fluid would possess at the same point on the body, by a correction factor. The factor is an infinite series of first derivatives of Legendre functions of the first kind and odd order. The first three coefficients in this series are computed for bodies of certain axis ratios, and graphs of the values of these coefficients against axis ratio are plotted. The behaviour of the nth coefficient for large values of n is given. Results for slender ellipsoids, considering these as a limiting case of the family of ellipsoids just referred to, are obtained and are found to agree with the usual slender-body theory. Using these an attempt is made to continue the graphs of the first three coefficients in the correction factor series for the whole range of axis ratios of the ellipsoids in the system, namely zero to unity. The results obtained for the bluff-nosed ellipsoids may be used to estimate the effects of compressibility on the pressure distribution over the front of a general bluff-nosed body in steady flow.

scholarly journals A study of the interactions between uniform and pointwise vortices in an inviscid fluid

2016 ◽  
Vol 7 (1) ◽  
pp. 4-22 ◽  
Author(s):  
Giorgio Riccardi
Keyword(s):  
Vortex Motion ◽  
Vortex Pair ◽  
Integral Relation ◽  
Successive Approximations ◽  
Inviscid Fluid ◽  
Schwarz Function ◽  
Strongly Nonlinear ◽  
Novel Approach ◽  
Dynamics Simulations ◽  
Turn Over

AbstractThe planar interactions between pair of vortices in an inviscid fluid are analytically investigated, by assuming one of the two vortices pointwise and the other one uniform. A novel approach using the Schwarz function of the boundary of the uniform vortex is adopted. It is based on a new integral relation between the (complex) velocity induced by the uniform vortex and its Schwarz function and on the time evolution equation of this function. They lead to a singular integrodifferential problem. Even if this problem is strongly nonlinear, its nonlinearities are confined inside two terms, only. As a consequence, its solution can be analytically approached by means of successive approximations. The ones at 0th (nonlinear terms neglected) and 1st (nonlinear terms evaluated on the 0-order solution) orders are calculated and compared with contour dynamics simulations of the vortex motion. A satisfactory agreement is keept for times which are small with respect to the turn-over time of the vortex pair.

Shape Effects on Viscous Damping and Motion of Heaving Cylinders1

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Ronald W. Yeung ◽  
Yichen Jiang
Keyword(s):  
Fluid Motion ◽  
Vortex Method ◽  
The Body ◽  
Floating Body ◽  
Navier Stokes ◽  
Inviscid Fluid ◽  
Fluid Viscosity ◽  
Viscous Damping ◽  
Viscous Effects ◽  
Shape Effects

Fluid viscosity is known to influence hydrodynamic forces on a floating body in motion, particularly when the motion amplitude is large and the body is of bluff shape. While traditionally these hydrodynamic force or force coefficients have been predicted by inviscid-fluid theory, much recent advances had taken place in the inclusion of viscous effects. Sophisticated Reynolds-Averaged Navier–Stokes (RANS) software are increasingly popular. However, they are often too elaborate for a systematic study of various parameters, geometry or frequency, where many runs with extensive data grid generation are needed. The Free-Surface Random-Vortex Method (FSRVM) developed at UC Berkeley in the early 2000 offers a middle-ground alternative, by which the viscous-fluid motion can be modeled by allowing vorticity generation be either turned on or turned off. The heavily validated FSRVM methodology is applied in this paper to examine how the draft-to-beam ratio and the shaping details of two-dimensional cylinders can alter the added inertia and viscous damping properties. A collection of four shapes is studied, varying from rectangles with sharp bilge corners to a reversed-curvature wedge shape. For these shapes, basic hydrodynamic properties are examined, with the effects of viscosity considered. With the use of these hydrodynamic coefficients, the motion response of the cylinders in waves is also investigated. The sources of viscous damping are clarified.

A family of steady, translating vortex pairs with distributed vorticity

1980 ◽  
Vol 99 (1) ◽  
pp. 129-144 ◽  
Author(s):  
R. T. Pierrehumbert
Keyword(s):  
Steady State ◽  
Vortex Pair ◽  
Relaxation Method ◽  
Numerical Results ◽  
Inviscid Fluid ◽  
Approximate Theory ◽  
Experimental Production ◽  
Vortex Pairs ◽  
The Family ◽  
Axis Of Symmetry

An efficient relaxation method is developed for computing the properties of a family of vortex pairs with distributed vorticity, propagating without change of shape through a homogeneous, inviscid fluid. The numerical results indicate that a steady state exists even when the gap between vortices is arbitrarily small, and that as the gap closes the steady state approaches a limiting vortex pair with a cusp on the axis of symmetry. Comparison is made with an approximate theory due to Saffman, and agreement is found to be good until the vortices are almost touching. The energy of members of the family is computed, and possible means of experimental production are discussed.

scholarly journals Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid

2000 ◽  
Vol 418 ◽  
pp. 167-188 ◽  
Author(s):  
PAUL BILLANT ◽  
JEAN-MARC CHOMAZ
Keyword(s):  
Turbulent Flows ◽  
Three Dimensional ◽  
Vortex Pair ◽  
Reynolds Numbers ◽  
Stratified Fluid ◽  
Viscous Effects ◽  
Homogeneous Fluid ◽  
Cross Sectional ◽  
Columnar Vortex ◽  
New Type

This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 < Fh0 < 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 > 0.21, the elliptic instability develops resulting in three-dimensional motions which eventually collapse into a relaminarized vortex pair. Irregular zigzags are then also observed to grow. The threshold for the inhibition of the elliptic instability Fh0 = 0.2±0.01 is independent of N and in good agreement with the theoretical study of Miyazaki & Fukumoto (1992). Complete stabilization for Fh0 < 0.13 is probably due to viscous effects since the associated Reynolds number is low, Re0 < 260. In geophysical flows characterized by low Froude numbers and large Reynolds numbers, we conjecture that this viscous stabilization will occur at much lower Froude number.It is tentatively argued that this new type of instability may explain the layering widely observed in stratified turbulent flows.

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